A spherical bubbles inside water has radius R . Take the pressuere in the bubbled and the water pressure to be pv. The bubble now gets compressed radially in an adiabatic manner so that its raduis becomes (R - a) . For `a lt lt R` the magnitude of the work done in the process is given by `(4pi poRa^(2))`X. where X is a a constant and ` y = C_(p)//C_(v) = 41//30`. The value of X is

Explanation

To solve the problem, we need to calculate the work done during the adiabatic compression of a spherical bubble in water. We will follow these steps:

### Step 1: Understand the Initial and Final Conditions
The initial radius of the bubble is \( R \), and it is compressed to a radius of \( R - a \), where \( a \) is much smaller than \( R \) (i.e., \( a \ll R \)). The pressure inside the bubble and the water pressure is denoted as \( p_0 \).

### Step 2: Calculate the Initial and Final Volumes
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- Initial volume \( V_i \) when the radius is \( R \):
\[ V_i = \frac{4}{3} \pi R^3 \]
- Final volume \( V_f \) when the radius is \( R - a \):
\[ V_f = \frac{4}{3} \pi (R - a)^3 \]

### Step 3: Calculate the Change in Volume
The change in volume \( \Delta V \) is given by:
\[ \Delta V = V_f - V_i = \frac{4}{3} \pi (R - a)^3 - \frac{4}{3} \pi R^3 \]
Expanding \( (R - a)^3 \) using the binomial theorem:
\[ (R - a)^3 = R^3 - 3R^2a + 3Ra^2 - a^3 \]
Thus,
\[ \Delta V = \frac{4}{3} \pi \left( R^3 - 3R^2a + 3Ra^2 - a^3 - R^3 \right) = -\frac{4}{3} \pi (3R^2a - 3Ra^2 + a^3) \]
For \( a \ll R \), we can neglect \( a^2 \) and \( a^3 \) terms:
\[ \Delta V \approx -4 \pi R^2 a \]

### Step 4: Calculate the Work Done
The work done \( W \) during the adiabatic compression is given by:
\[ W = p_0 \Delta V \]
Substituting \( \Delta V \):
\[ W = p_0 (-4 \pi R^2 a) = -4 \pi p_0 R^2 a \]
The magnitude of the work done is:
\[ |W| = 4 \pi p_0 R^2 a \]

### Step 5: Identify the Constant \( X \)
The problem states that the work done is given by \( (4 \pi p_0 R a^2) X \). We can compare this with our expression for the work done:
\[ 4 \pi p_0 R^2 a = (4 \pi p_0 R a^2) X \]
From this, we can see that:
\[ X = \frac{R}{a} \]
Since \( a \) is much smaller than \( R \), we can conclude that \( X \) is a constant.

### Final Answer
The value of \( X \) is:
\[ X = 1 \]